A shoutcaster is someone who is a commentator on a video game. The history of the shoutcaster began when games were played in game arcades and there would be a knowledgeable and entertaining person doing commentary behind the players.
From there games moved to TV and so the esports commentators used a proprietary software called SHOUTcast which allowed them to take the vision of the game and broadcast a commentary over the top of it. The software used tags and responses that all started with ICY standing for ‘I can yell’ so there is some truth in the popular etymology that shoutcasting comes from the need to yell over the noise of the game.
The style of commentating developed so that it has become standard to have two different styles of shoutcasting, the PBP (play-by-play) caster who follows the action, using their discretion to pick significant moves and add energy and narrative to the game story, and the colourcaster, usually an experienced and respected gamer, who can do analysis of the play. Both these styles are borrowed from existing sports commentary where one commentator does the detail and the other adds colour.
Finally games and commentary have moved to the internet where viewers can watch a game being played live with live commentary from the shoutcasters.